Experiments 24

Cut on the Bias

In 1878 Pafnuty Chebyshev told a congress in Paris how to cut cloth for a ball, then handed round a rubber ball he had dressed in two pieces. Woven cloth cannot stretch. Its threads can only shear, so every curved body turns the checks into diamonds until the threads jam. Pin gingham to a ball, a saddle and a bagel, and cut the pattern a tailor would need.

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Chebyshev’s ball, 1878: two pieces, one per half, with the threads running corner to corner across each pattern.

Drag to turn it · tap to pin the next pieceSwipe sideways to turn it · tap to pin more

Cutting table · laid flat

Covered
100%
Pieces
2
fewest found fewest 1
Cloth
1.08×
Known cuts
Notes on how it works

Notes

A talk about clothes

On 28 August 1878 Pafnuty Chebyshev, better known for his polynomials, gave a talk in Paris called “Sur la coupe des vêtements”, on the cutting of clothes. Two years earlier he had heard Édouard Lucas, the inventor of the Tower of Hanoi, speak on the geometry of weaving, and he wanted to say how to cut cloth for “a body of any shape”.

His draft fills nineteen pages of a school exercise book. It came with five cardboard cut-outs and a rubber ball he had dressed in two pieces of cloth, which went round the room and, by one account, ended up in a game in a school yard. A week later Gaston Darboux wrote to Jules Houël: “It is very original, but very ingenious.” Chebyshev never marked the manuscript for printing, so it stayed unpublished in French until Étienne Ghys printed a transcription in 2011.

Squares that lean

His starting point fits in a sentence. Wrap a body in cloth with small checks and “the rectangles turn into parallelograms whose angles depart more or less from ninety degrees, but the length of the sides does not change appreciably.” Woven cloth cannot stretch along its threads. All it can do is let the crossings scissor.

He added that two threads through one point, at right angles and along the shortest paths on the body, decide where every other thread goes. That is the pin and the grain here. The page then lays the cloth the way he would with a compass: each new crossing is the point one thread length from its two neighbors, row after row, up to 29,241 crossings for a piece on the ball.

Why a ball leans the checks

Walk around any patch of cloth on a curved body and add up its four corner angles. On a flat table they make four right angles. On a ball they fall short, and the shortfall is exactly how much of the ball the patch covers. This is Gauss’s curvature, read off the cloth. J. N. Hazzidakis wrote it down for these nets in 1880, and the tests behind this page check it to two decimal places.

A ball closes the checks one way and a saddle the other. A bagel does both: its outside curves like a ball, its hole like a saddle, and the two cancel exactly. A cylinder or a cone does neither, which is why sleeves and lampshades are easy. On a ball the checks lean fastest along the diagonals, so a pattern ends up with its corners on the threads and its edges cut on the bias, as in Chebyshev’s own two pieces.

One piece, 2011

Chebyshev covered half the ball with each piece. Ghys followed the same net further, “48.0151% further than Chebyshev, to be precise”, and showed that one piece cut like an envelope dresses the whole ball. The seam is two arcs crossing at the bottom, and at four points the threads lie flat on top of each other.

The angle between the threads obeys the sine-Gordon equation, a famous one in physics, and on Chebyshev’s net it depends only on the product of the two thread lengths from the pin. It first falls to zero at 1.8616 for a ball of radius one. The cloth here folds at the same place, which is one of the checks that the compass is doing the right thing.

Thread has width

Chebyshev’s threads have no thickness, so they can lean until they lie flat. Real yarns touch first. Threads of width w spaced s apart touch when the checks lean by arccos(w/s), and past that point the cloth wrinkles instead of following the body. Composites engineers call it the locking angle. The thread slider sets w/s, and the small swatch shows the gaps closing.

With threads that lock at 37°, Chebyshev’s two halves leave about a tenth of the ball bare, along the diagonals where the checks lean fastest. Turn one half 45° and its threads cover the other’s diagonals: two pieces close again. The two halves of a tennis ball interlock the same way, though felt is not woven and can stretch.

Where the seams go

Every piece is smoothed out from its pin at the same pace, and a seam forms where two meet. The pace counts lean as well as length: a stretch of cloth whose checks lean hard costs up to twice as much, because a tailor would rather use cloth that sits square. With that rule one piece on the ball finds Ghys’s seam without being told where it is.

Covered is measured at 8,000 points spread evenly over the surface. Cloth is the area of flat cloth used per area of surface covered. Leaning checks pack their threads closer, so a ball always takes more cloth than its own skin: 1.08 times for Chebyshev’s halves.

Out of whole cloth

Most accounts say Chebyshev took this up because a clothing firm hired him to save cloth on uniforms for the Crimean War. The story goes back to a 1996 post on the MacTutor history site with no sources. In 2025 Victor Tapia checked it against every biography and record he could find and concluded that it was “a fabrication, invented out of whole cloth.” Chebyshev himself credits Lucas’s talk.

Who cuts cloth like this now

Carbon fiber parts are laid up on molds with the same model. Draping software asks for a seed point and one warp and one weft direction through it, then lays the net out from there, exactly as Chebyshev proposed. Frei Otto’s 1975 timber gridshell in Mannheim is a Chebyshev net of wooden laths, bent from flat. In 1936 the engineer Popov, translating the manuscript into Russian, applied it to airship envelopes, kites and parachutes.

What it leaves out

The cloth has no weight, no friction and no stiffness against bending, and its threads do not stretch at all. Real yarns stretch a little, and tailors use that, along with darts and ease, to get past what pure geometry allows. The forms are exact shapes: a ball of radius one, a saddle z = (x² − y²)/2 cut off at radius 1.25, and a bagel whose tube is half its ring. The fewest-piece counts are the best a search found, not proofs that nothing better exists.

Sources